SDP backdating query

Client was living with his brother, Cl on ESA and MRC, brother on LRC. His brother’s DLA was then revised so that he was receiving MRC and they were both entitled to SDP.

Brother has then passed away suddenly shortly after. Client is only now requesting SDP some months later

In terms of backdating the SDP my reading is that it can be backdated to the date of change in the brother’s DLA. Had this not happened backdating would not have been possible as client would have become entitled to SDP through a change in circumstances rather than award of a qualifying benefit.

Have I got this right? It’s Friday and my brain is soup - client is v vulnerable and I want to be 100% sure before I raise his expectations of a lump sum payment